Aircraft: SMALL AIRCRAFT; HIGH WING; 1 ENG; FIXED GEAR
Phase of flight: Final Approach
Anomaly: Conflict NMAC; Deviation - Altitude Excursion From Assigned Altitude; Deviation / Discrepancy - Procedural FAR; Deviation / Discrepancy - Procedural Published Material / Policy

Vectoring student onto RNAV XX into ZZZ at 3;000 feet; 11nm South of the field; we switch to CTAF to start monitoring incoming traffic. On an estimated 030 heading giving the final vector to intercept for the RNAVXX; we hear our first transmission from the King Air stating '15nm South descending out of 5;000 feet inbound runway XX; ZZZ'. I tried to find the traffic visually and on TCAS but we could not visually see them as they were almost fully behind us and TCAS did not show any traffic in the area they reported; so I immediately made a radio call on CTAF declaring; 'ZZZ traffic; Aircraft X; 11nm South; intercepting the RNAV XX; 3;000 feet descending to 2;600 feet; King Air traffic you can overtake us before we come in.' The King Air advises the traffic call and states; 'We have someone on TCAS; that should be you.' With two-way communication established we continued with our approach descending to 2;600 feet expecting the King Air to sidestep and overtake us. After a few seconds of no communication; the King Air appears on our TCAS showing plus 2;000 feet about 13nm South of the field on an extended final for runway XX. As the traffic was behind us and high there was no way for me to visually identify the oncoming aircraft. As I couldn't maintain visual separation myself; I stated on CTAF 'We would appreciate 1;000ft of clearance while you overtake.' We received no response from the King Air. The TCAS altitude separation continued to reduce from 2;000 feet to 1;000 feet. As the traffic came within 1nm; altitude separation was under 1;000 feet and continued to rapidly reduce; I had my student began a descent to try and avoid a potential collision as it appeared the King Air was going to continue descending directly into our reported altitude. As the traffic passed overhead about 7nm from the end of the runway; the last TCAS message noted in the cockpit was +500ft and the traffic alerting system sounding from the G1000. During the descent to gain separation; we had to maneuver down to 1;900ft just to maintain a +500ft separation as the King Air continued to descend directly on top of us with no indication of leveling before crossing overhead. When I first gained a visual of the aircraft; it appeared the separation was much lower than the +500 feet last seen on the TCAS and estimated between +300/+400 feet. After the traffic was no longer within NMAC distance; I stated on CTAF; 'You could have given us some more clearance'. The King Air responds with; 'We gave you 500ft; learn to fly your airplane.' If he had 500ft of separation it was because we descended 700ft to avoid him. Multiple other CTAF transmissions were made between us after; fighting between who was at fault and who wasn't following the regulations. After landing; another crew that was on an approximate 1nm final for runway 36 stated that the separation on their TCAS as the King Air overtook us was +300 feet. The whole incident from first radio contact to final radio transmission between the two crews was an estimated 3-4 minutes. No injuries during the incident. There are several witnesses besides the two aircraft that were involved that are willing to corroborate the situation; their information will be available upon request. Flight Crew reported evasive action to avoid collision with aircraft that did not communicate their intentions clearly.